The family said that the NCAA acted hastily and without regard for due process and that it accepted the results of the school's internal investigation without further review.
The report from former FBI Director Louis Freeh said Paterno and three school officials concealed allegations against Sandusky dating to 1998.
Paterno's family and the three officials have denied the conclusions.
Sandusky awaits sentencing; he was convicted in June on 45 counts of sexually abusing boys.
The NCAA announced the sanctions on July 23. Among the penalties was a $60 million fine.
The school accepted the sanctions and signed off on a consent decree. University president Rodney Erickson said he saw no better option because the threat of the "death penalty" - shutting down the football program for at least a year - was hanging over the school.
A Penn State spokesman declined to comment Friday on the Paterno family's intent to appeal.
The Paternos' lawyer said the family had a right to appeal because it was named in the NCAA's consent decree and the Freeh report. The family said it hoped to formally submit an appeal and requested oral arguments before the NCAA's infractions appeal committee, its executive committee, or other leaders.
Sollers wrote that "the NCAA and Penn State's board [of trustees] chair and president entirely ignored the fact that the Freeh report, on which these extraordinary penalties are based, is deeply flawed because it is incomplete, rife with unsupported opinions, and unquestionably one-sided."
He called the sanctions possibly the most important disciplinary action in the history of the NCAA and said the organization had handled the situation in a fundamentally inappropriate and unprecedented manner.
Michael McCann, director of the Sports Law Institute and a professor at Vermont Law School, said Friday he doubted that the Paternos had standing to appeal.
"He's not alive, and his family itself would not seem to have any legal standing to challenge the NCAA," McCann said, "at least in terms of filing an appeal."
Associated Press writer Mark Scolforo in Harrisburg contributed to this article.
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